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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.catmull_rom"></a><a class="link" href="catmull_rom.html" title="Catmull-Rom Splines">Catmull-Rom Splines</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.catmull_rom.h0"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.synopsis"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.synopsis">Synopsis</a>
    </h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">catmull_rom</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

    <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Point</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Point</span><span class="special">&gt;</span> <span class="special">&gt;</span>
    <span class="keyword">class</span> <span class="identifier">catmull_rom</span>
    <span class="special">{</span>
    <span class="keyword">public</span><span class="special">:</span>

        <span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">points</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">closed</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">2</span><span class="special">)</span>

        <span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">Point</span><span class="special">&gt;</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">closed</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">)</span> <span class="number">2</span><span class="special">);</span>

        <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">s</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>

        <span class="identifier">Real</span> <span class="identifier">max_parameter</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>

        <span class="identifier">Real</span> <span class="identifier">parameter_at_point</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>

        <span class="identifier">Point</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">s</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
    <span class="special">};</span>

<span class="special">}}</span>
</pre>
<h4>
<a name="math_toolkit.catmull_rom.h1"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.description"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.description">Description</a>
    </h4>
<p>
      Catmull-Rom splines are a family of interpolating curves which are commonly
      used in computer graphics and animation. Catmull-Rom splines enjoy the following
      properties:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Affine invariance: The interpolant commutes with affine transformations.
        </li>
<li class="listitem">
          Local support of the basis functions: This gives stability and fast evaluation.
        </li>
<li class="listitem">
          <span class="emphasis"><em>C</em></span><sup>2</sup>-smoothness
        </li>
<li class="listitem">
          Interpolation of control points-this means the curve passes through the
          control points. Many curves (such as Bézier) are <span class="emphasis"><em>approximating</em></span>
          - they do not pass through their control points. This makes them more difficult
          to use than interpolating splines.
        </li>
</ul></div>
<p>
      The <code class="computeroutput"><span class="identifier">catmull_rom</span></code> class provided
      by Boost.Math creates a cubic Catmull-Rom spline from an array of points in
      any dimension. Since there are numerous ways to represent a point in <span class="emphasis"><em>n</em></span>-dimensional
      space, the class attempts to be flexible by templating on the point type. The
      requirements on the point type are discussing in more detail below, but roughly,
      it must have a dereference operator defined (e.g., <code class="computeroutput"><span class="identifier">p</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>
      is not a syntax error), it must be able to be dereferenced up to <code class="computeroutput"><span class="identifier">dimension</span> <span class="special">-</span><span class="number">1</span></code>, and <code class="computeroutput"><span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span></code>
      is of type <code class="computeroutput"><span class="identifier">Real</span></code>, define <code class="computeroutput"><span class="identifier">value_type</span></code>, and the free function <code class="computeroutput"><span class="identifier">size</span><span class="special">()</span></code>. These
      requirements are met by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
      and <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>. The basic usage is shown here:
    </p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">points</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">));</span>
<span class="comment">// Interpolate at s = 0.1:</span>
<span class="keyword">auto</span> <span class="identifier">point</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
</pre>
<p>
      Here, <span class="emphasis"><em>s</em></span> varies between 0 and <span class="emphasis"><em>cr.max_parameter()</em></span>.
      The latter's value depends on the choice of <span class="emphasis"><em>alpha</em></span>. In
      order to interpolate the curve between /points[i]/ and /points[i+1]<span class="emphasis"><em>,
      /s</em></span> should go from <span class="emphasis"><em>cr.parameter_at_point(i)</em></span>
      and <span class="emphasis"><em>cr.parameter_at_point(i+1)</em></span>.
    </p>
<p>
      The spline can be either open or <span class="emphasis"><em>closed</em></span>, closed meaning
      that there is some <span class="emphasis"><em>s &gt; 0</em></span> such that <span class="emphasis"><em>P(s) =
      P(0)</em></span>. The default is open, but this can be easily changed:
    </p>
<pre class="programlisting"><span class="comment">// closed = true</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">),</span> <span class="keyword">true</span><span class="special">);</span>
</pre>
<p>
      In either case, evaluating the interpolator at <span class="emphasis"><em>s=0</em></span> returns
      the first point in the list.
    </p>
<p>
      If the curve is open, then the first and last segments may have strange behavior.
      The traditional solution is to prepend a carefully selected control point to
      the data so that the first data segment (second interpolator segment) has reasonable
      tangent vectors, and simply ignore the first interpolator segment. A control
      point is appended to the data using similar criteria. However, we recommend
      not going through this effort until it proves to be necessary: For most use-cases,
      the curve is good enough without prepending and appending control points, and
      responsible selection of non-data control points is difficult.
    </p>
<p>
      Inside <code class="computeroutput"><span class="identifier">catmull_rom</span></code>, the curve
      is represented as closed. This is because an open Catmull-Rom curve is <span class="emphasis"><em>implicitly
      closed</em></span>, but the closing point is the zero vector. There is no reason
      to suppose that the zero vector is a better closing point than the endpoint
      (or any other point, for that matter), so traditionally Catmull-Rom splines
      leave the segment between the first and second point undefined, as well as
      the segment between the second-to-last and last point. We find this property
      of the traditional implementation of Catmull-Rom splines annoying and confusing
      to the user. Hence internally, we close the curve so that the first and last
      segments are defined. Of course, this causes the <span class="emphasis"><em>tangent</em></span>
      vectors to the first and last points to be bizarre. This is a "pick your
      poison" design decision-either the curve cannot interpolate in its first
      and last segments, or the tangents along the first and last segments are meaningless.
      In the vast majority of cases, this will be no problem to the user. However,
      if it becomes a problem, then the user should add one extra point in a position
      they believe is reasonable and close the curve.
    </p>
<p>
      Since the routine internally represents the curve as closed, a question arises:
      Why does the user have to specify if the curve is open or closed? The answer
      is that the parameterization is chosen by the routine, so it is of interest
      to the user to understand the values where a meaningful result is returned.
    </p>
<pre class="programlisting"><span class="identifier">Real</span> <span class="identifier">max_s</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">.</span><span class="identifier">max_parameter</span><span class="special">();</span>
</pre>
<p>
      If you attempt to interpolate for <code class="computeroutput"><span class="identifier">s</span>
      <span class="special">&gt;</span> <span class="identifier">max_s</span></code>,
      an exception is thrown. If the curve is closed, then <code class="computeroutput"><span class="identifier">cr</span><span class="special">(</span><span class="identifier">max_s</span><span class="special">)</span>
      <span class="special">=</span> <span class="identifier">p0</span></code>,
      where <code class="computeroutput"><span class="identifier">p0</span></code> is the first point
      on the curve. If the curve is open, then <code class="computeroutput"><span class="identifier">cr</span><span class="special">(</span><span class="identifier">max_s</span><span class="special">)</span>
      <span class="special">=</span> <span class="identifier">pf</span></code>,
      where <code class="computeroutput"><span class="identifier">pf</span></code> is the final point
      on the curve.
    </p>
<p>
      The Catmull-Rom curve admits an infinite number of parameterizations. The default
      parameterization of the <code class="computeroutput"><span class="identifier">catmull_rom</span></code>
      class is the so-called <span class="emphasis"><em>centripedal</em></span> parameterization. This
      parameterization has been shown to be the only parameterization that does not
      form cusps or self-intersections within segments. However, for advanced users,
      other parameterizations can be chosen using the <span class="emphasis"><em>alpha</em></span>
      parameter:
    </p>
<pre class="programlisting"><span class="comment">// alpha = 1 is the "chordal" parameterization.</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">),</span> <span class="keyword">false</span><span class="special">,</span> <span class="number">1.0</span><span class="special">);</span>
</pre>
<p>
      The alpha parameter must always be in the range <code class="computeroutput"><span class="special">[</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">]</span></code>.
    </p>
<p>
      Finally, the tangent vector to any point of the curve can be computed via
    </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">s</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
<span class="identifier">Point</span> <span class="identifier">tangent</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">s</span><span class="special">);</span>
</pre>
<p>
      Since the magnitude of the tangent vector is dependent on the parameterization,
      it is not meaningful (unless the user chooses the chordal parameterization
      <span class="emphasis"><em>alpha = 1</em></span> which parameterizes by Euclidean distance between
      points.) However, its direction is meaningful no matter the parameterization,
      so the user may wish to normalize this result.
    </p>
<h4>
<a name="math_toolkit.catmull_rom.h2"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.examples"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.examples">Examples</a>
    </h4>
<h4>
<a name="math_toolkit.catmull_rom.h3"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.performance"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.performance">Performance</a>
    </h4>
<p>
      The following performance numbers were generated for a call to the Catmull-Rom
      interpolation method. The number that follows the slash is the number of points
      passed to the interpolant. We see that evaluation of the interpolant is 𝑶(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)).
    </p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="number">2700</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
  <span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
  <span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
  <span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">262</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
  <span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">3145</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">Benchmark</span>                              <span class="identifier">Time</span>           <span class="identifier">CPU</span>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4</span>               <span class="number">20</span> <span class="identifier">ns</span>         <span class="number">20</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">21</span> <span class="identifier">ns</span>         <span class="number">21</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">23</span> <span class="identifier">ns</span>         <span class="number">23</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">24</span> <span class="identifier">ns</span>         <span class="number">24</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">27</span> <span class="identifier">ns</span>         <span class="number">27</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">27</span> <span class="identifier">ns</span>         <span class="number">27</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">30</span> <span class="identifier">ns</span>         <span class="number">30</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>             <span class="number">32</span> <span class="identifier">ns</span>         <span class="number">31</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>            <span class="number">33</span> <span class="identifier">ns</span>         <span class="number">33</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>            <span class="number">34</span> <span class="identifier">ns</span>         <span class="number">34</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>            <span class="number">36</span> <span class="identifier">ns</span>         <span class="number">36</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>            <span class="number">38</span> <span class="identifier">ns</span>         <span class="number">38</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>           <span class="number">39</span> <span class="identifier">ns</span>         <span class="number">39</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>           <span class="number">40</span> <span class="identifier">ns</span>         <span class="number">40</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>           <span class="number">45</span> <span class="identifier">ns</span>         <span class="number">44</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>          <span class="number">46</span> <span class="identifier">ns</span>         <span class="number">46</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>          <span class="number">50</span> <span class="identifier">ns</span>         <span class="number">50</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>          <span class="number">53</span> <span class="identifier">ns</span>         <span class="number">52</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>         <span class="number">58</span> <span class="identifier">ns</span>         <span class="number">57</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>          <span class="number">2.97</span> <span class="identifier">lgN</span>       <span class="number">2.97</span> <span class="identifier">lgN</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>             <span class="number">19</span> <span class="special">%</span>         <span class="number">19</span> <span class="special">%</span>
</pre>
<h4>
<a name="math_toolkit.catmull_rom.h4"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.point_types"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.point_types">Point
      types</a>
    </h4>
<p>
      We have already discussed that certain conditions on the <code class="computeroutput"><span class="identifier">Point</span></code>
      type template argument must be obeyed. The following shows a custom point type
      in 3D which can be used as a template argument to Catmull-Rom:
    </p>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">mypoint3d</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
    <span class="comment">// Must define a value_type:</span>
    <span class="keyword">typedef</span> <span class="identifier">Real</span> <span class="identifier">value_type</span><span class="special">;</span>

    <span class="comment">// Regular constructor--need not be of this form.</span>
    <span class="identifier">mypoint3d</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">{</span><span class="identifier">m_vec</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">;</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">y</span><span class="special">;</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">z</span><span class="special">;</span> <span class="special">}</span>

    <span class="comment">// Must define a default constructor:</span>
    <span class="identifier">mypoint3d</span><span class="special">()</span> <span class="special">{}</span>

    <span class="comment">// Must define array access:</span>
    <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span> <span class="keyword">const</span>
    <span class="special">{</span>
        <span class="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
    <span class="special">}</span>

    <span class="comment">// Must define array element assignment:</span>
    <span class="identifier">Real</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span>
    <span class="special">{</span>
        <span class="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
    <span class="special">}</span>

<span class="keyword">private</span><span class="special">:</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;</span> <span class="identifier">m_vec</span><span class="special">;</span>
<span class="special">};</span>


<span class="comment">// Must define the free function "size()":</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<span class="keyword">constexpr</span> <span class="identifier">size_t</span> <span class="identifier">size</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">)</span>
<span class="special">{</span>
    <span class="keyword">return</span> <span class="number">3</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
      These conditions are satisfied by both <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code> and
      <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>, but it may nonetheless be useful
      to define your own point class so that (say) you can define geometric distance
      between them.
    </p>
<h4>
<a name="math_toolkit.catmull_rom.h5"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.caveats"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.caveats">Caveats</a>
    </h4>
<p>
      The Catmull-Rom interpolator requires memory for three more points than is
      provided by the user. This causes the class to call a <code class="computeroutput"><span class="identifier">resize</span><span class="special">()</span></code> on the input vector. If <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">capacity</span><span class="special">()</span> <span class="special">&gt;=</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
      <span class="special">+</span> <span class="number">3</span></code>,
      then no problems arise; there are no reallocs, and in practice this condition
      is almost always satisfied. However, if <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">capacity</span><span class="special">()</span> <span class="special">&lt;</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
      <span class="special">+</span> <span class="number">3</span></code>,
      the <code class="computeroutput"><span class="identifier">realloc</span></code> causes a performance
      penalty of roughly 20%.
    </p>
<h4>
<a name="math_toolkit.catmull_rom.h6"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.generic_containers"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.generic_containers">Generic
      Containers</a>
    </h4>
<p>
      The <code class="computeroutput"><span class="identifier">Point</span></code> type may be stored
      in a different container than <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>.
      For example, here is how to store the points in a Boost.uBLAS vector:
    </p>
<pre class="programlisting"><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p0</span><span class="special">(</span><span class="number">0.1</span><span class="special">,</span> <span class="number">0.2</span><span class="special">,</span> <span class="number">0.3</span><span class="special">);</span>
<span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p1</span><span class="special">(</span><span class="number">0.2</span><span class="special">,</span> <span class="number">0.3</span><span class="special">,</span> <span class="number">0.4</span><span class="special">);</span>
<span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p2</span><span class="special">(</span><span class="number">0.3</span><span class="special">,</span> <span class="number">0.4</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
<span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p3</span><span class="special">(</span><span class="number">0.4</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.6</span><span class="special">);</span>
<span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p4</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="number">0.6</span><span class="special">,</span> <span class="number">0.7</span><span class="special">);</span>
<span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">p5</span><span class="special">(</span><span class="number">0.6</span><span class="special">,</span> <span class="number">0.7</span><span class="special">,</span> <span class="number">0.8</span><span class="special">);</span>

<span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&gt;</span> <span class="identifier">u</span><span class="special">(</span><span class="number">6</span><span class="special">);</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p0</span><span class="special">;</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p1</span><span class="special">;</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p2</span><span class="special">;</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p3</span><span class="special">;</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">4</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p4</span><span class="special">;</span>
<span class="identifier">u</span><span class="special">[</span><span class="number">5</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p5</span><span class="special">;</span>

<span class="comment">// Tests initializer_list:</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">u</span><span class="special">)&gt;</span> <span class="identifier">cat</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">u</span><span class="special">));</span>
</pre>
<h4>
<a name="math_toolkit.catmull_rom.h7"></a>
      <span class="phrase"><a name="math_toolkit.catmull_rom.references"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.references">References</a>
    </h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Cem Yuksel, Scott Schaefer, and John Keyser, <span class="emphasis"><em>Parameterization
          and applications of Catmull–Rom curves</em></span>, Computer-Aided Design
          43 (2011) 747–755.
        </li>
<li class="listitem">
          Phillip J. Barry and Ronald N. Goldman, <span class="emphasis"><em>A Recursive Evaluation
          Algorithm for a Class of Catmull-Rom Splines</em></span>, Computer Graphics,
          Volume 22, Number 4, August 1988
        </li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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